Kate Middleton -- who was admitted to London's King Edward VII Hospital on Dec. 3 for acute morning sickness -- is already on the road to recovery, St. James's Palace confirmed to Us Weekly on Tuesday, Dec. 4.

"The Duchess of Cambridge is continuing to feel better," a rep for the royal couple said in a statement. "She and the Duke are immensely grateful for the good wishes they have received. She will remain in hospital at present and will continue to be treated for hyperemesis gravidarum."

Hyperemesis gravidarum is a rare condition that can cause severe nausea, vomiting, weight loss, dehydration, lightheadedness and fainting. Middleton, 30, is among the less than 1% of mothers-to-be who suffer from it. Because of that -- and because she is still in the very early stages of her pregnancy -- doctors at King Edward VII will be monitoring her health closely for at least the next several days, after which time she'll be released and require a period of rest, the palace told Us.

In the meantime, the Duchess can lean on her husband for support. Earlier today, Prince William was spotted returning to the hospital to be by his wife's side. Already a doting dad-to-be, he arrived before noon and stayed several hours, leaving just a couple of minutes before 6 p.m.